Mathematics
Grade 10
15 min
Introduction to similar solids
Introduction to similar solids
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Introduction & Learning Objectives
Learning Objectives
Define the properties of similar solids and identify if two solids are similar.
Determine the scale factor between two similar solids using their corresponding linear dimensions.
Articulate the relationship between the linear scale factor, the ratio of surface areas, and the ratio of volumes.
Use the scale factor to calculate the unknown surface area of a similar solid.
Use the scale factor to calculate the unknown volume of a similar solid.
Solve multi-step problems by working backwards from the ratio of volumes or surface areas to find the linear scale factor.
Ever wonder how a tiny, detailed model car can be a perfect replica of a real one? 🚗 It's all about the mathematics of similar solids!
In this lesson, we'll explore the relationship be...
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Key Concepts & Vocabulary
TermDefinitionExample
Similar SolidsTwo solids of the same type are similar if the ratio of their corresponding linear measures (such as height, radius, or edge length) is constant. This means they have the exact same shape, but can be different sizes.A basketball with a 12 cm radius and a tennis ball with a 3.3 cm radius are similar solids (spheres). A rectangular prism of 2x3x4 is similar to one of 4x6x8.
Corresponding PartsThe matching sides, edges, radii, heights, or other linear measurements on two similar solids.In two similar cones, the radius of the base of the first cone corresponds to the radius of the base of the second cone. The height of the first corresponds to the height of the second.
Scale Factor (k)The constant ratio of the lengths of corresponding parts of two similar s...
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Core Formulas
Ratio of Surface Areas
If the scale factor of two similar solids is a:b, then the ratio of their corresponding surface areas is a²:b².
Use this rule when you know the linear scale factor and need to find a missing surface area, or vice-versa. Remember that area is a two-dimensional measurement, so the scale factor is squared.
Ratio of Volumes
If the scale factor of two similar solids is a:b, then the ratio of their corresponding volumes is a³:b³.
Use this rule when you know the linear scale factor and need to find a missing volume, or vice-versa. Volume is a three-dimensional measurement, so the scale factor is cubed.
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Challenging
A large cone is cut by a plane parallel to its base, one-third of the way down from the vertex. This creates a smaller, similar cone on top. What is the ratio of the volume of the small top cone to the volume of the remaining bottom part (the frustum)?
A.1:26
B.1:27
C.1:8
D.1:9
Challenging
The height of a cylinder is increased by 10% to create a new, larger cylinder that is similar to the original. By what percentage does the volume of the cylinder increase?
A.10%
B.21%
C.30%
D.33.1%
Challenging
The volumes of two similar solids are 54 m³ and 128 m³. The surface area of the smaller solid is 90 m². Find the surface area of the larger solid.
A.160 m²
B.213.3 m²
C.120 m²
D.144 m²
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